Procesado de Señales e Imágenes Médicas

Ingeniería Biomédica

Ph.D. Pablo Eduardo Caicedo Rodríguez

2024-01-22

Part I: Probability and Statistics (Repaso)

Events, Sample Space, Experiments

Definition

An experiment is a physical procedure that produces some kind of result.

Definition

An event is a set of experiment’s possible results.

Consejo

A sample space is the set of ALL possibles results of an experiment.

data = np.genfromtxt(path_ecg+"/mitbih_train.csv", delimiter=",")
ecg1 = data[1, :-1]
time = np.array(range(0,len(ecg1)))/125
fig = plt.figure()
plt.plot(time, ecg1)
plt.xlabel("Time (s)")
plt.ylabel("Normalized ECG")

print("Maximun Value: "+ str(ecg1.max()))

Maximun Value: 1.0

print("Minimun Value: "+ str(ecg1.min()))

Minimun Value: 0.0

print(ecg1[np.random.choice(ecg1.shape[0], 1, replace=False)])

[0.]

Name: ECG Heartbeat Categorization Dataset.

URL: https://www.kaggle.com/datasets/shayanfazeli/heartbeat?resource=download

Probability Axioms

For the given events A and B that are in a sample space S:

Axioms

\(0 \leq P_r \left(A\right) \leq 1\)
\(P_r\left(S\right) = 1\)
If \(A \cap B = \emptyset\) then \(P_r\left(A \cup B \right) = P_r \left(A\right) + P_r \left(B\right)\)
If \(A \cap B \neq \emptyset\) then \(P_r\left(A \cup B \right) = P_r \left(A\right) + P_r \left(B\right) - P_r\left(A \cap B \right)\)
\(P_r\left(\bar{A}\right) = 1-P_r \left(A\right)\)
If \(A\subset B\) then \(P_r \left(A\right)\leq P_r \left(B\right)\)
\(P_r \left(A|B\right)=\frac{P_r \left(A\cap B\right)}{P_r \left(B\right)}\)

Random Variable

Definition

A random variable is a real valued function of the elements of a sample space, S . Given an experiment, E , with sample space, S, the random variable maps each possible outcome of E.

Definition

The probability mass function (PMF), \(P_X\left(x\right)\), of a random variable, X, is a function that assigns a probability to each possible value of the random variable, X.

Random Variable

Random Variables

Conditions

Discrete

\[\sum_{\chi \in X}P_X\left(\chi \right) = 1\]

Continuous

\[\int_{-\infty}^{\infty}P_X\left(\chi \right)d\chi = 1\]

Random Variables

Expected Values

Discrete

\[\mu = \sum_{\chi \in X}\chi P_X\left(\chi \right)\]

Continuous

\[\mu=\int_{-\infty}^{\infty}\chi P_X\left(\chi \right)d\chi = 1\]

Random Variables

Variance

Discrete

\[\sigma^2 = \sum_{\chi \in X}\left(\chi - \mu \right)^2 P_X\left(\chi \right)\]

Continuous

\[\sigma^2 = \int_{-\infty}^{\infty}\left(\chi - \mu \right)^2 P_X\left(\chi \right)d\chi = 1\]

PDF Estimation

Graph
Code
Exp. Value
Variance

counts01, bin_edges01 = np.histogram(ecg1, bins=10, density=True)
counts02, bin_edges02 = np.histogram(ecg1, bins=50, density=True)
counts03, bin_edges03 = np.histogram(ecg1, bins=100, density=True)
fig01=plt.figure()
plt.plot(bin_edges01[1:], counts01/sum(counts01), label="Estimation with 10 bins")
plt.plot(bin_edges02[1:], counts02/sum(counts02), label="Estimation with 50 bins")
plt.plot(bin_edges03[1:], counts03/sum(counts03), label="Estimation with 100 bins")
plt.legend()
plt.grid()
plt.xlabel("Normalised ECG Value")
plt.ylabel("Estimated PDF Value")

0.09001020772910533

0.02551116143316462